IndisputableMonolith.Measurement.TwoBranchGeodesic
This module supplies the geometric definitions for a two-branch geodesic rotation in recognition paths, starting at angle θ_s and terminating at π/2. Downstream bridge modules cite it to establish the exact C=2A relation between recognition cost and residual action. The module consists entirely of supporting definitions and invariants with no internal proofs.
claimA two-branch geodesic rotation in recognition path space from initial angle $θ_s$ to terminal angle $π/2$, equipped with residual action $A$ and rate action functions that satisfy the pointwise kernel identity $J(r(θ)) = 2 tan θ$.
background
The module imports the lightweight PathAction interface, which defines recognition paths together with their action and weight functionals while omitting heavy measure-theoretic results. It introduces the two-branch rotation as the central object for quantum measurement, along with sibling definitions for residual action, rate action, amplitude squares, and normalization invariants. These constructs operate in the setting where recognition cost C is computed along geodesics on the phi-ladder.
proof idea
This is a definition module containing no proofs. It exports the rotation object and its derived action functions for direct use in downstream equivalence statements.
why it matters in Recognition Science
The definitions feed the central C=2A theorem in C2ABridge and the kernel-matching lemma in KernelMatch, which together establish the constructive identity between recognition cost and residual-model rate action. The module supplies the geometric component required for the measurement bridge in the Recognition framework.
scope and limits
- Does not contain measure-theoretic lemmas on path additivity or domain shifts.
- Does not prove the C=2A equivalence or kernel identity.
- Does not treat multi-branch or higher-dimensional rotations.
- Does not address continuous limits or integral forms of the action.